↑

Properties

Label 2-336-112.61-c2-0-31
Degree $2$
Conductor $336$
Sign $0.963 - 0.269i$
Analytic cond. $9.15533$
Root an. cond. $3.02577$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.729 + 1.86i)2-s + (−1.67 − 0.448i)3-s + (−2.93 − 2.71i)4-s + (5.66 − 1.51i)5-s + (2.05 − 2.78i)6-s + (−2.28 + 6.61i)7-s + (7.20 − 3.48i)8-s + (2.59 + 1.50i)9-s + (−1.30 + 11.6i)10-s + (−18.4 − 4.93i)11-s + (3.69 + 5.86i)12-s + (12.6 − 12.6i)13-s + (−10.6 − 9.08i)14-s − 10.1·15-s + (1.23 + 15.9i)16-s + (16.1 − 9.33i)17-s + ⋯
L(s)  = 1  + (−0.364 + 0.931i)2-s + (−0.557 − 0.149i)3-s + (−0.733 − 0.679i)4-s + (1.13 − 0.303i)5-s + (0.342 − 0.464i)6-s + (−0.326 + 0.945i)7-s + (0.900 − 0.435i)8-s + (0.288 + 0.166i)9-s + (−0.130 + 1.16i)10-s + (−1.67 − 0.448i)11-s + (0.307 + 0.488i)12-s + (0.976 − 0.976i)13-s + (−0.761 − 0.648i)14-s − 0.677·15-s + (0.0772 + 0.997i)16-s + (0.951 − 0.549i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.963 - 0.269i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.963 - 0.269i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $0.963 - 0.269i$
Analytic conductor: \(9.15533\)
Root analytic conductor: \(3.02577\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{336} (61, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 336,\ (\ :1),\ 0.963 - 0.269i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.16904 + 0.160326i\)
\(L(\frac12)\) \(\approx\) \(1.16904 + 0.160326i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.729 - 1.86i)T \)
3 \( 1 + (1.67 + 0.448i)T \)
7 \( 1 + (2.28 - 6.61i)T \)
good5 \( 1 + (-5.66 + 1.51i)T + (21.6 - 12.5i)T^{2} \)
11 \( 1 + (18.4 + 4.93i)T + (104. + 60.5i)T^{2} \)
13 \( 1 + (-12.6 + 12.6i)T - 169iT^{2} \)
17 \( 1 + (-16.1 + 9.33i)T + (144.5 - 250. i)T^{2} \)
19 \( 1 + (3.59 + 13.4i)T + (-312. + 180.5i)T^{2} \)
23 \( 1 + (-27.4 - 15.8i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + (-1.09 - 1.09i)T + 841iT^{2} \)
31 \( 1 + (-52.0 + 30.0i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 + (-4.13 - 15.4i)T + (-1.18e3 + 684.5i)T^{2} \)
41 \( 1 - 8.66T + 1.68e3T^{2} \)
43 \( 1 + (-16.2 + 16.2i)T - 1.84e3iT^{2} \)
47 \( 1 + (-56.7 - 32.7i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (-45.8 - 12.2i)T + (2.43e3 + 1.40e3i)T^{2} \)
59 \( 1 + (-4.66 + 17.4i)T + (-3.01e3 - 1.74e3i)T^{2} \)
61 \( 1 + (10.8 + 40.6i)T + (-3.22e3 + 1.86e3i)T^{2} \)
67 \( 1 + (-10.1 + 38.0i)T + (-3.88e3 - 2.24e3i)T^{2} \)
71 \( 1 - 29.6iT - 5.04e3T^{2} \)
73 \( 1 + (1.79 + 3.10i)T + (-2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (-24.9 + 43.1i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (4.13 - 4.13i)T - 6.88e3iT^{2} \)
89 \( 1 + (30.2 - 52.3i)T + (-3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + 42.9iT - 9.40e3T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.11179850215401033677684320972, −10.23441195468552339545414340843, −9.471860123441156126947121435649, −8.501794185029870287261155147639, −7.60756509481924481462506275207, −6.22357878416312577234765583707, −5.58621036653227805846063115357, −5.11187361997900583775339532937, −2.75638788918412335179131865046, −0.826690202357654295125347990019, 1.17860031388645919139001621774, 2.62748856961647068208382337391, 4.02597545616899895241867265294, 5.21528765934692967645853393294, 6.40255764032163623375505701302, 7.56031596425443641208990111321, 8.741465617323329465168714033426, 9.997342506735161347281878590960, 10.32618721923469398260083068513, 10.90103403761339729379595004809

Graph of the $Z$-function along the critical line